3.20 \(\int e^{2 x} x \, dx\)

Optimal. Leaf size=20 \[ \frac{1}{2} e^{2 x} x-\frac{e^{2 x}}{4} \]

[Out]

-E^(2*x)/4 + (E^(2*x)*x)/2

_______________________________________________________________________________________

Rubi [A]  time = 0.0144578, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{1}{2} e^{2 x} x-\frac{e^{2 x}}{4} \]

Antiderivative was successfully verified.

[In]  Int[E^(2*x)*x,x]

[Out]

-E^(2*x)/4 + (E^(2*x)*x)/2

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.13688, size = 14, normalized size = 0.7 \[ \frac{x e^{2 x}}{2} - \frac{e^{2 x}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(2*x)*x,x)

[Out]

x*exp(2*x)/2 - exp(2*x)/4

_______________________________________________________________________________________

Mathematica [A]  time = 0.00214549, size = 15, normalized size = 0.75 \[ e^{2 x} \left (\frac{x}{2}-\frac{1}{4}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[E^(2*x)*x,x]

[Out]

E^(2*x)*(-1/4 + x/2)

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 12, normalized size = 0.6 \[{\frac{ \left ( 2\,x-1 \right ){{\rm e}^{2\,x}}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(2*x)*x,x)

[Out]

1/4*(2*x-1)*exp(2*x)

_______________________________________________________________________________________

Maxima [A]  time = 1.35264, size = 15, normalized size = 0.75 \[ \frac{1}{4} \,{\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*e^(2*x),x, algorithm="maxima")

[Out]

1/4*(2*x - 1)*e^(2*x)

_______________________________________________________________________________________

Fricas [A]  time = 0.2061, size = 15, normalized size = 0.75 \[ \frac{1}{4} \,{\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*e^(2*x),x, algorithm="fricas")

[Out]

1/4*(2*x - 1)*e^(2*x)

_______________________________________________________________________________________

Sympy [A]  time = 0.064042, size = 10, normalized size = 0.5 \[ \frac{\left (2 x - 1\right ) e^{2 x}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(2*x)*x,x)

[Out]

(2*x - 1)*exp(2*x)/4

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.20213, size = 15, normalized size = 0.75 \[ \frac{1}{4} \,{\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*e^(2*x),x, algorithm="giac")

[Out]

1/4*(2*x - 1)*e^(2*x)